Finite Group Databasebeta

← #16857245

Subgroups — #16857245

Order 3968.

510 conjugacy classes of subgroups.

The subgroup lattice has 430 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11758493C211yesmaximal, characteristic, complemented
2#14521718C211yesmaximal, characteristic, complemented
3#14521693C211yesmaximal, characteristic, complemented
4#14521437C211yesmaximal, characteristic, complemented
5#14521295C211yesmaximal, characteristic, complemented
6#14521282C211yesmaximal, characteristic, complemented
7#14521269C211yesmaximal, characteristic, complemented
8#14521242C211yesmaximal, characteristic, complemented
9#14521211C211yesmaximal, characteristic, complemented
10#14521193C211yesmaximal, characteristic, complemented
11#14521191C211yesmaximal, characteristic, complemented
12#14521183C211yesmaximal, characteristic, complemented
13#14521150C211yesmaximal, characteristic, complemented
14#14521118C211yesmaximal, characteristic, complemented
15C2×C31×AGL(1,ℤ/8)C211yesmaximal, characteristic, complemented
16C2×#41×C31C2211yescharacteristic
17C22×D4×C31C2211yescharacteristic
18C2×#92499C2211yescharacteristic, complemented
19C2×#92497C2211yescharacteristic
20C31×AGL(1,ℤ/8)—24no—
21C2×SD16×C31C2211yescharacteristic
22C2×SD16×C31C2211yescharacteristic
23C2×D8×C31C2211yescharacteristic
24C2×D8×C31C2211yescharacteristic
25C2×M16×C31C2211yescharacteristic
26#11758530—21no—
27C2×C22⋊Dic31—21no—
28Q8×Dic31C2211yescharacteristic, complemented
29C2×#92482C2211yescharacteristic, complemented
30#11758517—21no—
31#11758517C2211yescharacteristic, complemented
32#11758516—21no—
33#11758515—21no—
34#11758515C2211yescharacteristic
35#11758515C2211yescharacteristic, complemented
36#11758513C2211yescharacteristic
37#11758512—21no—
38D4×Dic31C2211yescharacteristic
39C2×#92480C2211yescharacteristic, complemented
40C2×#92479C2211yescharacteristic
41C2×#92479C2211yescharacteristic
42#11758507—21no—
43#11758507C2211yescharacteristic
44C4×C22⋊D31—21no—
45C4×C22⋊D31C2211yescharacteristic
46#11758501C2211yescharacteristic
47#11758493—24no—
48C2×#92470C2211yescharacteristic, complemented
49C2×C8×D31C2211yescharacteristic, complemented
50#11758423C2211yescharacteristic, complemented
51#11758420C2211yescharacteristic
52#11758420C2211yescharacteristic
53#11758420C2211yescharacteristic, complemented
54C22⋊(C31⋊C8)C2211yescharacteristic
55#11758409C2211yescharacteristic, complemented
56(C2×D31)⋊C8C2211yescharacteristic, complemented
57#11758406C2211yescharacteristic, complemented
58#11758404C2211yescharacteristic, complemented
59#11758403C2211yescharacteristic, complemented
60Dic31⋊C8C2211yescharacteristic, complemented
61C24×C31—21no—
62#41×C31—22no—
63#41×C31—41no—
64C2×Q8×C31C2311yescharacteristic
65C2×D4×C31—22no—
66C2×D4×C31—21no—
67C2×D4×C31—21no—
68C2×D4×C31—41no—
69C2×D4×C31C2311yescharacteristic
70C2×D4×C31C2311yescharacteristic
71C2×D4×C31C2311yescharacteristic
72C22×C4×C31—21no—
73C22×C4×C31C2311yescharacteristic
74C2×C22⋊D31—21no—
75C2×C22⋊D31—21no—
76C2×C22⋊D31—21no—
77C2×C22⋊D31—21no—
78C2×C22⋊D31—21no—
79C2×C22⋊D31—21no—
80C2×C22⋊D31D411yescharacteristic
81C22×Dic31—21no—
82C22×Dic31—21no—
83#92499—22no—
84#92499—41no—
85#92497—22no—
86#92497—41no—
87C2×D124—21no—
88C2×D124C2311yescharacteristic
89C2×C4×D31—21no—
90C2×C4×D31C2311yescharacteristic
91C2×Dic62C2311yescharacteristic
92SD16×C31—22no—
93SD16×C31D414yescomplemented
94D8×C31—22no—
95D8×C31—22no—
96M16×C31—22no—
97C2×C8×C31C2311yescharacteristic
98C2×C8×C31C2311yescharacteristic
99C22⋊Dic31—21no—
100C22⋊Dic31—21no—
101C22⋊Dic31—21no—
102C22⋊Dic31—21no—
103C22⋊Dic31—21no—
104C22⋊Dic31—21no—
105C22⋊Dic31D411yescharacteristic
106#92482—22no—
107#92480—41no—
108#92479—22no—
109#92479—41no—
110(C2×D31)⋊C4—21no—
111(C2×D31)⋊C4D411yescharacteristic
112C4⋊Dic31—21no—
113C4⋊Dic31C2311yescharacteristic
114C4⋊Dic31C2311yescharacteristic
115Dic31⋊C4—21no—
116Dic31⋊C4D411yescharacteristic
117C4×Dic31—21no—
118C4×Dic31C2311yescharacteristic
119C2×C31⋊C8C2311yescharacteristic
120C2×C31⋊C8C2311yescharacteristic
121#92470—22no—
122C8×D31—22no—
123C23×C31—21no—
124C23×C31—21no—
125C23×C31—21no—
126C23×C31—42no—
127C23×C31—42no—
128C23×C31—81no—
129C23×C31C2×D411yescharacteristic
130C22×D31—21no—
131C22×D31C2×D411yescharacteristic
132Q8×C31—21no—
133Q8×C31C2×D412yes—
134D4×C31—21no—
135D4×C31—21no—
136D4×C31—21no—
137D4×C31—21no—
138D4×C31—21no—
139D4×C31—42no—
140D4×C31—42no—
141D4×C31C2×D412yes—
142C2×C4×C31—21no—
143C2×C4×C31—21no—
144C2×C4×C31—21no—
145C2×C4×C31—42no—
146C2×C4×C31C2×D411yescharacteristic
147C2×C4×C31C2411yescharacteristic
148C22⋊D31—41no—
149C22⋊D31—42no—
150C22⋊D31—42no—
151C22⋊D31—81no—
152C22⋊D31—81no—
153C22⋊D31—81no—
154C22⋊D31—81no—
155C2×Dic31—21no—
156C2×Dic31—21no—
157C2×Dic31—21no—
158C2×Dic31—42no—
159C2×Dic31—81no—
160C2×Dic31C2×D411yescharacteristic
161C2×Dic31C2×D411yescharacteristic
162C2×Dic31C2×D411yescharacteristic
163D124—21no—
164D124—21no—
165D124—42no—
166C4×D31—21no—
167C4×D31—21no—
168C4×D31—42no—
169Dic62—21no—
170Dic62—21no—
171C8×C31—21no—
172C8×C31C2×D412yes—
173C31⋊C8—21no—
174C31⋊C8—21no—
175#3268—311nomaximal, Sylow
176C22×C31—22no—
177C22×C31—21no—
178C22×C31—21no—
179C22×C31—21no—
180C22×C31—41no—
181C22×C31—41no—
182C22×C31—41no—
183C22×C31—41no—
184C22×C31—41no—
185C22×C31—41no—
186C22×C31—41no—
187C22×C31—41no—
188C22×C31—84no—
189C22×C31C22×D411yescharacteristic
190C2×D31—21no—
191C2×D31—21no—
192C2×D31—21no—
193C2×D31—41no—
194C2×D31—41no—
195C2×D31—41no—
196C4×C31—22no—
197C4×C31—21no—
198C4×C31C22×D411yescharacteristic
199C4×C31H(5,𝔽2)11yescharacteristic
200Dic31—21no—
201Dic31—21no—
202Dic31—21no—
203Dic31—41no—
204Dic31—41no—
205Dic31—41no—
206C2×AGL(1,ℤ/8)D3111yescharacteristic, complemented
207C2×#135—311no—
208#550—311no—
209#546—311no—
210#496—311no—
211#468—311no—
212#466—311no—
213#464—311no—
214#459—311no—
215#452—311no—
216#449—311no—
217#449—311no—
218#447—311no—
219#440—311no—
220#435—311no—
221C2×C31—41no—
222C2×C31—42no—
223C2×C31—81no—
224C2×C31—81no—
225C2×C31D4211yescharacteristic
226C2×C31#57511yescharacteristic
227C2×C31#57611yescharacteristic
228D31—41no—
229D31—81no—
230C2×#41—311no—
231C2×#41—311no—
232C2×#41C2×D3111yescharacteristic
233C22×D4C2×D3111yescharacteristic
234AGL(1,ℤ/8)—24no—
235#135—624no—
236C2×Q16—311no—
237C2×SD16—311no—
238C2×SD16—311no—
239C2×SD16C2×D3111yescharacteristic
240C2×SD16C2×D3111yescharacteristic
241C2×D8—311no—
242C2×D8—311no—
243C2×D8C2×D3111yescharacteristic
244C2×D8C2×D3111yescharacteristic
245C2×M16C2×D3111yescharacteristic
246C22×C8—311no—
247#127—311no—
248#127—621no—
249#124—311no—
250#123—621no—
251C22⋊Q8—311no—
252#121—311no—
253#121—311no—
254#121—311no—
255#121—621no—
256#121—621no—
257#121—621no—
258#120—621no—
259C4×Q8—311no—
260C4×D4—311no—
261C4×D4—311no—
262C4×D4—621no—
263C2×C22⋊C4—621no—
264#107—311no—
265C4⋊C8—311no—
266#103—311no—
267#103—311no—
268#102—311no—
269#102—311no—
270#102—311no—
271#102—311no—
272C22⋊C8—311no—
273C22⋊C8—311no—
274#97—311no—
275C31#326811yescharacteristic, complemented, Sylow
276C24—21no—
277#41—22no—
278#41—41no—
279#41—622no—
280#41—622no—
281#41—1241no—
282#41—1241no—
283C2×Q8—311no—
284C2×Q8C22×D3111yescharacteristic
285C2×D4—22no—
286C2×D4—21no—
287C2×D4—21no—
288C2×D4—41no—
289C2×D4—311no—
290C2×D4—311no—
291C2×D4—621no—
292C2×D4—621no—
293C2×D4—621no—
294C2×D4—621no—
295C2×D4—621no—
296C2×D4—621no—
297C2×D4—621no—
298C2×D4C22×D3111yescharacteristic
299C2×D4C22×D3111yescharacteristic
300C2×D4C22×D3111yescharacteristic
301C22×C4—21no—
302C22×C4—311no—
303C22×C4—621no—
304C22×C4—621no—
305C22×C4—621no—
306C22×C4C22×D3111yescharacteristic
307Q16—622no—
308SD16—22no—
309SD16—622no—
310SD16—1241no—
311SD16C22⋊D3114yescomplemented
312D8—22no—
313D8—22no—
314D8—622no—
315D8—1241no—
316M16—22no—
317C2×C8—311no—
318C2×C8—311no—
319C2×C8—622no—
320C2×C8C22×D3111yescharacteristic
321C2×C8C22×D3111yescharacteristic
322C4⋊C4—311no—
323C4⋊C4—311no—
324C4⋊C4—311no—
325C4⋊C4—621no—
326C4⋊C4—621no—
327C22⋊C4—311no—
328C22⋊C4—311no—
329C22⋊C4—621no—
330C22⋊C4—621no—
331C22⋊C4—621no—
332C22⋊C4—621no—
333C22⋊C4—621no—
334C22⋊C4—621no—
335C22⋊C4—621no—
336C42—311no—
337C42—621no—
338C23—21no—
339C23—21no—
340C23—21no—
341C23—42no—
342C23—42no—
343C23—81no—
344C23—311no—
345C23—621no—
346C23D4×D3111yescharacteristic
347Q8—21no—
348Q8—621no—
349Q8—621no—
350Q8C2×C22⋊D3112yes—
351D4—21no—
352D4—21no—
353D4—21no—
354D4—21no—
355D4—21no—
356D4—42no—
357D4—42no—
358D4—621no—
359D4—621no—
360D4—1242no—
361D4—1241no—
362D4—1242no—
363D4—1242no—
364D4—2481no—
365D4—2481no—
366D4—2481no—
367D4—2481no—
368D4C2×C22⋊D3112yes—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—42no—
373C2×C4—311no—
374C2×C4—311no—
375C2×C4—311no—
376C2×C4—621no—
377C2×C4—621no—
378C2×C4—621no—
379C2×C4—621no—
380C2×C4—621no—
381C2×C4—1242no—
382C2×C4—1242no—
383C2×C4—2481no—
384C2×C4D4×D3111yescharacteristic
385C2×C4C23×D3111yescharacteristic
386C8—21no—
387C8—621no—
388C8—621no—
389C8C2×C22⋊D3112yes—
390C22—21no—
391C22—22no—
392C22—21no—
393C22—21no—
394C22—41no—
395C22—41no—
396C22—41no—
397C22—41no—
398C22—41no—
399C22—41no—
400C22—41no—
401C22—41no—
402C22—84no—
403C22—621no—
404C22—621no—
405C22—621no—
406C22—1241no—
407C22—1241no—
408C22—1241no—
409C22C2×D4×D3111yescharacteristic
410C4—21no—
411C4—22no—
412C4—621no—
413C4—621no—
414C4—621no—
415C4—1241no—
416C4—1241no—
417C4—1241no—
418C4#1175856111yescharacteristic
419C4C22×C22⋊D3111yescharacteristic
420C2—41no—
421C2—42no—
422C2—81no—
423C2—81no—
424C2—1241no—
425C2—2481no—
426C2D4×C22⋊D3111yescharacteristic
427C2#1452184511yescharacteristic
428C2#1452185311yescharacteristic
subgroupG/H
centerC22C2×D4×D31
commutator, cosocleC2×C4×C31C24
FrattiniC2×C4C23×D31
FittingC2×C31×AGL(1,ℤ/8)C2
socleC22×C31C22×D4
hypercenterC2×AGL(1,ℤ/8)D31
hypocenterC31#3268
LCSG ▷ C2×C4×C31 ▷ C2×C31 ▷ C31
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×AGL(1,ℤ/8)

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. C2×#11758493C2semidirect
2. #14521718C2semidirect
3. #14521693C2semidirect
4. #14521437C2semidirect
5. #14521295C2semidirect
6. #14521282C2semidirect
7. #14521269C2semidirect
8. #14521242C2semidirect
9. #14521211C2semidirect
10. #14521193C2semidirect
11. #14521191C2semidirect
12. #14521183C2semidirect
13. #14521150C2semidirect
14. #14521118C2semidirect
15. C2×C31×AGL(1,ℤ/8)C2semidirect
18. C2×#92499C22semidirect
28. Q8×Dic31C22semidirect
29. C2×#92482C22semidirect
31. #11758517C22semidirect
35. #11758515C22semidirect
39. C2×#92480C22semidirect
48. C2×#92470C22semidirect
49. C2×C8×D31C22semidirect
50. #11758423C22semidirect
53. #11758420C22semidirect
55. #11758409C22semidirect
56. (C2×D31)⋊C8C22semidirect
57. #11758406C22semidirect
58. #11758404C22semidirect
59. #11758403C22semidirect
60. Dic31⋊C8C22semidirect
93. SD16×C31D4semidirect
206. C2×AGL(1,ℤ/8)D31semidirect
275. C31#3268semidirect
311. SD16C22⋊D31semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×#117584932
#145217182
#145216932
#145214372
#145212952
#145212822
#145212692
#145212422
#145212112
#145211932
#145211912
#145211832
#145211502
#145211182
C2×C31×AGL(1,ℤ/8)2
#326831